Deal a pile of cards face down from a standard 52-card deck. What is the minimum number of cards the pile must contain to ensure that it has at least five cards of the same suit?

To ensure that you have at least five cards of the same suit in a pile from a standard 52-card deck, we can use the Pigeonhole Principle. The principle states that if you have n pigeonholes and more than n pigeons, at least one pigeonhole must contain more than one pigeon.

A standard deck of cards has four suits: hearts, diamonds, clubs, and spades. If you want at least five cards of the same suit, you have to consider the worst-case scenario in which the distribution of suits is as even as possible before a fifth card of the same suit appears.

So, you could deal out 4 cards of each suit (hearts, diamonds, clubs, spades) in the best-distributed manner, which is a total of 4 suits x 4 cards each = 16 cards. After this, the next card you deal, no matter the suit, would be the fifth card of that suit.

Therefore, you must deal a minimum of 16 cards (to evenly distribute four cards of each suit) + 1 (to ensure a fifth card of any one suit), which gives us a total of 17 cards to make sure that there are at least five cards of the same suit in the pile.